MATH 590: Meshfree Methods

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1 MATH 590: Meshfree Methods Chapter 14: The Power Function and Native Space Error Estimates Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 MATH 590 Chapter 14 1

2 Outline 1 Fill Distance and Approximation Orders 2 Lagrange Form of the Interpolant and Cardinal Basis Functions 3 The Power Function 4 Generic Error Estimates for Functions in N K (Ω) 5 Error Estimates in Terms of the Fill Distance fasshauer@iit.edu MATH 590 Chapter 14 2

3 Outline Fill Distance and Approximation Orders 1 Fill Distance and Approximation Orders 2 Lagrange Form of the Interpolant and Cardinal Basis Functions 3 The Power Function 4 Generic Error Estimates for Functions in N K (Ω) 5 Error Estimates in Terms of the Fill Distance fasshauer@iit.edu MATH 590 Chapter 14 3

4 Fill Distance and Approximation Orders Goal: to provide error estimates for scattered data interpolation with strictly (conditionally) positive definite functions. MATH 590 Chapter 14 4

5 Fill Distance and Approximation Orders Goal: to provide error estimates for scattered data interpolation with strictly (conditionally) positive definite functions. We will provide most of the details for the strictly positive definite case, and only mention the extension to the conditionally positive definite case in the end. MATH 590 Chapter 14 4

6 Fill Distance and Approximation Orders Goal: to provide error estimates for scattered data interpolation with strictly (conditionally) positive definite functions. We will provide most of the details for the strictly positive definite case, and only mention the extension to the conditionally positive definite case in the end. In their final form we will want our estimates to depend on some kind of measure of the data distribution. MATH 590 Chapter 14 4

7 Fill Distance and Approximation Orders Goal: to provide error estimates for scattered data interpolation with strictly (conditionally) positive definite functions. We will provide most of the details for the strictly positive definite case, and only mention the extension to the conditionally positive definite case in the end. In their final form we will want our estimates to depend on some kind of measure of the data distribution. The measure that is usually used in approximation theory is the so-called fill distance already introduced in Chapter 2. h = h X,Ω = sup min x x j 2 x j X x Ω fasshauer@iit.edu MATH 590 Chapter 14 4

8 Fill Distance and Approximation Orders The fill distance indicates how well the data fill out the domain Ω. It denotes the radius of the largest empty ball that can be placed among the data locations. MATH 590 Chapter 14 5

9 Fill Distance and Approximation Orders The fill distance indicates how well the data fill out the domain Ω. It denotes the radius of the largest empty ball that can be placed among the data locations. Convergence We will be interested in whether the error f P (h) f tends to zero as h 0, and if so, how fast. fasshauer@iit.edu MATH 590 Chapter 14 5

10 Fill Distance and Approximation Orders The fill distance indicates how well the data fill out the domain Ω. It denotes the radius of the largest empty ball that can be placed among the data locations. Convergence We will be interested in whether the error f P (h) f tends to zero as h 0, and if so, how fast. Here {P (h) } h presents a sequence of interpolation (or, more generally, projection) operators that vary with the fill distance h. fasshauer@iit.edu MATH 590 Chapter 14 5

11 Fill Distance and Approximation Orders The fill distance indicates how well the data fill out the domain Ω. It denotes the radius of the largest empty ball that can be placed among the data locations. Convergence We will be interested in whether the error f P (h) f tends to zero as h 0, and if so, how fast. Here {P (h) } h presents a sequence of interpolation (or, more generally, projection) operators that vary with the fill distance h. Remark Most error bounds will focus on this worst-case setting. Some will be measured in the L 2 -norm, i.e., for average case errors, or other L p -norms. fasshauer@iit.edu MATH 590 Chapter 14 5

12 Fill Distance and Approximation Orders Example Let P (h) denote interpolation to data given at (2 n + 1) s, n = 1, 2,..., equally spaced points in the unit cube in R s so that h = 1 s (2 n +1) s 1 = 2 n. fasshauer@iit.edu MATH 590 Chapter 14 6

13 Fill Distance and Approximation Orders Example Let P (h) denote interpolation to data given at (2 n + 1) s, n = 1, 2,..., equally spaced points in the unit cube in R s so that h = 1 s (2 n +1) s 1 = 2 n. The definition of the fill distance also covers scattered data such as sets of Halton points. In fact, since Halton points are quasi-uniformly distributed (see Appendix A) we can assume h 2 n for a set of (2 n + 1) s Halton points in R s. fasshauer@iit.edu MATH 590 Chapter 14 6

14 Fill Distance and Approximation Orders Example Let P (h) denote interpolation to data given at (2 n + 1) s, n = 1, 2,..., equally spaced points in the unit cube in R s so that h = 1 s (2 n +1) s 1 = 2 n. The definition of the fill distance also covers scattered data such as sets of Halton points. In fact, since Halton points are quasi-uniformly distributed (see Appendix A) we can assume h 2 n for a set of (2 n + 1) s Halton points in R s. Remark These relations explain the specific sizes of the point sets we used in earlier examples. fasshauer@iit.edu MATH 590 Chapter 14 6

15 Fill Distance and Approximation Orders Since we want to employ the machinery of reproducing kernel Hilbert spaces presented in the previous chapter we will concentrate on error estimates for functions f N K. fasshauer@iit.edu MATH 590 Chapter 14 7

16 Fill Distance and Approximation Orders Since we want to employ the machinery of reproducing kernel Hilbert spaces presented in the previous chapter we will concentrate on error estimates for functions f N K. In the next chapter we will also mention some more general estimates. fasshauer@iit.edu MATH 590 Chapter 14 7

17 Fill Distance and Approximation Orders We measure the speed of convergence to zero in terms of approximation order. MATH 590 Chapter 14 8

18 Fill Distance and Approximation Orders We measure the speed of convergence to zero in terms of approximation order. We say that the approximation operator P (h) has L p -approximation order k if f P (h) f p = O(h k ) for h 0. fasshauer@iit.edu MATH 590 Chapter 14 8

19 Fill Distance and Approximation Orders We measure the speed of convergence to zero in terms of approximation order. We say that the approximation operator P (h) has L p -approximation order k if f P (h) f p = O(h k ) for h 0. Moreover, if we can also show that f P (h) f p o(h k ), then P (h) has exact L p -approximation order k. fasshauer@iit.edu MATH 590 Chapter 14 8

20 Fill Distance and Approximation Orders We measure the speed of convergence to zero in terms of approximation order. We say that the approximation operator P (h) has L p -approximation order k if f P (h) f p = O(h k ) for h 0. Moreover, if we can also show that f P (h) f p o(h k ), then P (h) has exact L p -approximation order k. Remark We will concentrate mostly on the case p = (i.e., pointwise estimates), but approximation order in other norms can also be studied. fasshauer@iit.edu MATH 590 Chapter 14 8

21 Fill Distance and Approximation Orders Remark In order to keep the following discussion as transparent as possible we will restrict ourselves to strictly positive definite functions. With (considerably) more technical details the following can also be formulated for strictly conditionally positive definite functions (see [Wendland (2005a)] for details). MATH 590 Chapter 14 9

22 Lagrange Form of the Interpolant and Cardinal Basis Functions Outline 1 Fill Distance and Approximation Orders 2 Lagrange Form of the Interpolant and Cardinal Basis Functions 3 The Power Function 4 Generic Error Estimates for Functions in N K (Ω) 5 Error Estimates in Terms of the Fill Distance fasshauer@iit.edu MATH 590 Chapter 14 10

23 Lagrange Form of the Interpolant and Cardinal Basis Functions The key idea for the following discussion is to express the interpolant in Lagrange form, i.e., using so-called cardinal basis functions. MATH 590 Chapter 14 11

24 Lagrange Form of the Interpolant and Cardinal Basis Functions The key idea for the following discussion is to express the interpolant in Lagrange form, i.e., using so-called cardinal basis functions. For radial basis function approximation this idea is due to [Wu and Schaback (1993)]. MATH 590 Chapter 14 11

25 Lagrange Form of the Interpolant and Cardinal Basis Functions The key idea for the following discussion is to express the interpolant in Lagrange form, i.e., using so-called cardinal basis functions. For radial basis function approximation this idea is due to [Wu and Schaback (1993)]. In the previous chapters we established that, for any strictly positive definite function Φ, the linear system Ac = y with A ij = Φ(x i x j ), i, j = 1,..., N, c = [c 1,..., c N ] T, and y = [f (x 1 ),..., f (x N )] T has a unique solution. fasshauer@iit.edu MATH 590 Chapter 14 11

26 Lagrange Form of the Interpolant and Cardinal Basis Functions In the following we will consider the more general situation where K is a strictly positive definite kernel, i.e., the entries of A are given by A ij = K (x i, x j ). fasshauer@iit.edu MATH 590 Chapter 14 12

27 Lagrange Form of the Interpolant and Cardinal Basis Functions In the following we will consider the more general situation where K is a strictly positive definite kernel, i.e., the entries of A are given by A ij = K (x i, x j ). The uniqueness result holds in this case also. fasshauer@iit.edu MATH 590 Chapter 14 12

28 Lagrange Form of the Interpolant and Cardinal Basis Functions In the following we will consider the more general situation where K is a strictly positive definite kernel, i.e., the entries of A are given by A ij = K (x i, x j ). The uniqueness result holds in this case also. In order to obtain the cardinal basis functions uj, j = 1,..., N, with the property uj (x i) = δ ij, i.e., u j (x i) = { 1 if i = j, 0 if i j, fasshauer@iit.edu MATH 590 Chapter 14 12

29 Lagrange Form of the Interpolant and Cardinal Basis Functions In the following we will consider the more general situation where K is a strictly positive definite kernel, i.e., the entries of A are given by A ij = K (x i, x j ). The uniqueness result holds in this case also. In order to obtain the cardinal basis functions uj, j = 1,..., N, with the property uj (x i) = δ ij, i.e., u j (x i) = we consider the linear system { 1 if i = j, 0 if i j, Au (x) = b(x), (1) where the matrix A is as above (and therefore invertible), u = [u 1,..., u N ]T, and b = [K (, x 1 ),..., K (, x N )] T. fasshauer@iit.edu MATH 590 Chapter 14 12

30 Lagrange Form of the Interpolant and Cardinal Basis Functions Existence of Cardinal Functions Theorem Suppose K is a strictly positive definite kernel on R s R s. Then, for any distinct points x 1,..., x N, there exist functions u j span{k (, x j ), j = 1,..., N} such that u j (x i) = δ ij. They are determined pointwise by solving the linear system (1), i.e., Au (x) = b(x). fasshauer@iit.edu MATH 590 Chapter 14 13

31 Lagrange Form of the Interpolant and Cardinal Basis Functions Existence of Cardinal Functions Theorem Suppose K is a strictly positive definite kernel on R s R s. Then, for any distinct points x 1,..., x N, there exist functions u j span{k (, x j ), j = 1,..., N} such that u j (x i) = δ ij. They are determined pointwise by solving the linear system (1), i.e., Au (x) = b(x). Therefore if we know the cardinal functions we can write the interpolant P f to f at x 1,..., x N in the cardinal form P f (x) = N f (x j )uj (x), x Rs. j=1 fasshauer@iit.edu MATH 590 Chapter 14 13

32 Lagrange Form of the Interpolant and Cardinal Basis Functions Remark Cardinal functions do not depend on the data values of the interpolation problem. MATH 590 Chapter 14 14

33 Lagrange Form of the Interpolant and Cardinal Basis Functions Remark Cardinal functions do not depend on the data values of the interpolation problem. They do heavily depend on the data locations (see plots on following slides). MATH 590 Chapter 14 14

34 Lagrange Form of the Interpolant and Cardinal Basis Functions Remark Cardinal functions do not depend on the data values of the interpolation problem. They do heavily depend on the data locations (see plots on following slides). Once the data sites are fixed and the basic function is chosen with an appropriate shape parameter (whose optimal value will depend on the data sites and values), then the cardinal functions are determined by the linear system (1). MATH 590 Chapter 14 14

35 Lagrange Form of the Interpolant and Cardinal Basis Functions Example Gaussian Cardinal Functions Figure: Cardinal functions for Gaussian interpolation (with ε = 5) on 81 uniformly gridded points in [0, 1] 2. Centered at an edge point (left) and at an interior point (right). fasshauer@iit.edu MATH 590 Chapter 14 15

36 Lagrange Form of the Interpolant and Cardinal Basis Functions Example More Gaussians Figure: Cardinal functions for Gaussian interpolation (with ε = 5) on 81 tensor-product Chebyshev points in [0, 1] 2. Centered at an edge point (left) and at an interior point (right). fasshauer@iit.edu MATH 590 Chapter 14 16

37 Lagrange Form of the Interpolant and Cardinal Basis Functions Example More Gaussians Figure: Cardinal functions for Gaussian interpolation (with ε = 5) on 81 Halton points in [0, 1] 2. Centered at an edge point (left) and at an interior point (right). fasshauer@iit.edu MATH 590 Chapter 14 17

38 Lagrange Form of the Interpolant and Cardinal Basis Functions Example Multiquadric Cardinal Functions Figure: Cardinal functions for multiquadric interpolation (with ε = 5) on 81 Halton points in [0, 1] 2. Centered at an edge point (left) and at an interior point (right). fasshauer@iit.edu MATH 590 Chapter 14 18

39 Lagrange Form of the Interpolant and Cardinal Basis Functions Remark Basic functions that grow with increasing distance from the center point (such as multiquadrics) are sometimes criticized for being counter-intuitive for scattered data approximation. MATH 590 Chapter 14 19

40 Lagrange Form of the Interpolant and Cardinal Basis Functions Remark Basic functions that grow with increasing distance from the center point (such as multiquadrics) are sometimes criticized for being counter-intuitive for scattered data approximation. The plot above shows that the associated cardinal functions are just as localized as those for the Gaussian basic functions, and thus the function space spanned by multiquadrics is a good local space. fasshauer@iit.edu MATH 590 Chapter 14 19

41 Lagrange Form of the Interpolant and Cardinal Basis Functions Program (RBFCardinalFunction.m) 1 rbf exp(-(e*r).^2); ep = 5; 2 N = 81; dsites = CreatePoints(N,2, u ); 3 ctrs = dsites; 4 neval = 40; M = neval^2; 5 epoints = CreatePoints(M,2, u ); 6 DM_data = DistanceMatrix(dsites,ctrs); 7 IM = rbf(ep,dm_data); % transpose of usual eval matrix 8 DM_B = DistanceMatrix(ctrs,epoints); 9 B = rbf(ep,dm_b); % many right-hand sides for (1) 10 cardfuns = IM\B; % one per ROW 11 xe = reshape(epoints(:,1),neval,neval); 12 ye = reshape(epoints(:,2),neval,neval); 13 CFplot=surf(xe,ye,reshape(cardfuns(50,:),neval,neval)); 14 set(cfplot, FaceColor, interp, EdgeColor, none ) 15 colormap autumn; view([145 45]); camlight 16 lighting gouraud Note that the code is different than in the book and avoids loops. fasshauer@iit.edu MATH 590 Chapter 14 20

42 Outline The Power Function 1 Fill Distance and Approximation Orders 2 Lagrange Form of the Interpolant and Cardinal Basis Functions 3 The Power Function 4 Generic Error Estimates for Functions in N K (Ω) 5 Error Estimates in Terms of the Fill Distance fasshauer@iit.edu MATH 590 Chapter 14 21

43 The Power Function Another important ingredient needed for our error estimates is the so-called power function. MATH 590 Chapter 14 22

44 The Power Function Another important ingredient needed for our error estimates is the so-called power function. To this end, we consider a domain Ω R s. Then for any strictly positive definite kernel K C(Ω Ω), any set of distinct points X = {x 1,..., x N } Ω, and an arbitrary vector u R N, we define the quadratic form N N N Q(u) = K (x, x) 2 u j K (x, x j ) + u i u j K (x i, x j ). j=1 i=1 j=1 fasshauer@iit.edu MATH 590 Chapter 14 22

45 The Power Function Another important ingredient needed for our error estimates is the so-called power function. To this end, we consider a domain Ω R s. Then for any strictly positive definite kernel K C(Ω Ω), any set of distinct points X = {x 1,..., x N } Ω, and an arbitrary vector u R N, we define the quadratic form N N N Q(u) = K (x, x) 2 u j K (x, x j ) + u i u j K (x i, x j ). Definition j=1 i=1 j=1 Suppose Ω R s and K C(Ω Ω) is strictly positive definite. For any distinct points X = {x 1,..., x N } Ω the power function P K,X is defined pointwise by [P K,X (x)] 2 = Q(u (x)), where u is the vector of cardinal functions studied above. fasshauer@iit.edu MATH 590 Chapter 14 22

46 The Power Function Using the definition of the native inner product norm from the previous chapter we can rewrite the quadratic form Q(u) as N N N Q(u) = K (x, x) 2 u j K (x, x j ) + u i u j K (x i, x j ) j=1 i=1 j=1 fasshauer@iit.edu MATH 590 Chapter 14 23

47 The Power Function Using the definition of the native inner product norm from the previous chapter we can rewrite the quadratic form Q(u) as N N N Q(u) = K (x, x) 2 u j K (x, x j ) + u i u j K (x i, x j ) j=1 = K (, x), K (, x) NK (Ω) 2 + N i=1 j=1 i=1 j=1 N u j K (, x), K (, x j ) NK (Ω) j=1 N u i u j K (, x i ), K (, x j ) NK (Ω) fasshauer@iit.edu MATH 590 Chapter 14 23

48 The Power Function Using the definition of the native inner product norm from the previous chapter we can rewrite the quadratic form Q(u) as N N N Q(u) = K (x, x) 2 u j K (x, x j ) + u i u j K (x i, x j ) j=1 = K (, x), K (, x) NK (Ω) 2 + N i=1 j=1 = K (, x) i=1 j=1 N u j K (, x), K (, x j ) NK (Ω) j=1 N u i u j K (, x i ), K (, x j ) NK (Ω) N u j K (, x j ), K (, x) j=1 N u j K (, x j ) NK (Ω) j=1 fasshauer@iit.edu MATH 590 Chapter 14 23

49 The Power Function Using the definition of the native inner product norm from the previous chapter we can rewrite the quadratic form Q(u) as N N N Q(u) = K (x, x) 2 u j K (x, x j ) + u i u j K (x i, x j ) j=1 = K (, x), K (, x) NK (Ω) 2 + N i=1 j=1 = K (, x) = K (, x) i=1 j=1 N u j K (, x), K (, x j ) NK (Ω) j=1 N u i u j K (, x i ), K (, x j ) NK (Ω) N u j K (, x j ), K (, x) j=1 N u j K (, x j ) j=1 2 N K (Ω) N u j K (, x j ) NK (Ω) j=1. (2) fasshauer@iit.edu MATH 590 Chapter 14 23

50 The Power Function Remark The name power function was chosen by [Schaback (1993)] based on its connection to the power function of a statistical decision function (originally introduced in [Neyman and Pearson (1936)]). fasshauer@iit.edu MATH 590 Chapter 14 24

51 The Power Function Remark The name power function was chosen by [Schaback (1993)] based on its connection to the power function of a statistical decision function (originally introduced in [Neyman and Pearson (1936)]). In the paper [Wu and Schaback (1993)] the power function was referred to as kriging function. This terminology comes from geostatistics (see, e.g., [Myers (1992)]). fasshauer@iit.edu MATH 590 Chapter 14 24

52 The Power Function Remark The name power function was chosen by [Schaback (1993)] based on its connection to the power function of a statistical decision function (originally introduced in [Neyman and Pearson (1936)]). In the paper [Wu and Schaback (1993)] the power function was referred to as kriging function. This terminology comes from geostatistics (see, e.g., [Myers (1992)]). In the statistics literature, the power function is known as the kriging variance (see, e.g., [Berlinet and Thomas-Agnan (2004), Matheron (1965), Stein (1999)]). fasshauer@iit.edu MATH 590 Chapter 14 24

53 The Power Function Using the linear system notation employed earlier, i.e., A ij = K (x i, x j ), i, j = 1,..., N, u = [u 1,..., u N ] T, and b = [K (, x 1 ),..., K (, x N )] T, we note that we can also rewrite the quadratic form Q(u) as Q(u) = K (x, x) 2 N u j K (x, x j ) + j=1 N i=1 j=1 N u i u j K (x i, x j ) fasshauer@iit.edu MATH 590 Chapter 14 25

54 The Power Function Using the linear system notation employed earlier, i.e., A ij = K (x i, x j ), i, j = 1,..., N, u = [u 1,..., u N ] T, and b = [K (, x 1 ),..., K (, x N )] T, we note that we can also rewrite the quadratic form Q(u) as Q(u) = K (x, x) 2 N u j K (x, x j ) + j=1 N i=1 j=1 N u i u j K (x i, x j ) = K (x, x) 2u T b(x) + u T Au. (3) fasshauer@iit.edu MATH 590 Chapter 14 25

55 The Power Function Using the linear system notation employed earlier, i.e., A ij = K (x i, x j ), i, j = 1,..., N, u = [u 1,..., u N ] T, and b = [K (, x 1 ),..., K (, x N )] T, we note that we can also rewrite the quadratic form Q(u) as Q(u) = K (x, x) 2 N u j K (x, x j ) + j=1 N i=1 j=1 N u i u j K (x i, x j ) = K (x, x) 2u T b(x) + u T Au. (3) This suggests two alternative representations of the power function. fasshauer@iit.edu MATH 590 Chapter 14 25

56 The Power Function Using the linear system notation employed earlier, i.e., A ij = K (x i, x j ), i, j = 1,..., N, u = [u 1,..., u N ] T, and b = [K (, x 1 ),..., K (, x N )] T, we note that we can also rewrite the quadratic form Q(u) as Q(u) = K (x, x) 2 N u j K (x, x j ) + j=1 N i=1 j=1 N u i u j K (x i, x j ) = K (x, x) 2u T b(x) + u T Au. (3) This suggests two alternative representations of the power function. Using the matrix-vector notation for Q(u), the power function is given as P K,X (x) = Q(u (x)) fasshauer@iit.edu MATH 590 Chapter 14 25

57 The Power Function Using the linear system notation employed earlier, i.e., A ij = K (x i, x j ), i, j = 1,..., N, u = [u 1,..., u N ] T, and b = [K (, x 1 ),..., K (, x N )] T, we note that we can also rewrite the quadratic form Q(u) as Q(u) = K (x, x) 2 N u j K (x, x j ) + j=1 N i=1 j=1 N u i u j K (x i, x j ) = K (x, x) 2u T b(x) + u T Au. (3) This suggests two alternative representations of the power function. Using the matrix-vector notation for Q(u), the power function is given as P K,X (x) = Q(u (x)) = K (x, x) 2(u (x)) T b(x) + (u (x)) T Au (x). fasshauer@iit.edu MATH 590 Chapter 14 25

58 The Power Function However, by the definition of the cardinal functions Au (x) = b(x), and therefore we have the two new variants P K,X (x) = K (x, x) 2(u (x)) T b(x) + (u (x)) T Au (x) fasshauer@iit.edu MATH 590 Chapter 14 26

59 The Power Function However, by the definition of the cardinal functions Au (x) = b(x), and therefore we have the two new variants P K,X (x) = K (x, x) 2(u (x)) T b(x) + (u (x)) T Au (x) = K (x, x) (u (x)) T b(x) fasshauer@iit.edu MATH 590 Chapter 14 26

60 The Power Function However, by the definition of the cardinal functions Au (x) = b(x), and therefore we have the two new variants P K,X (x) = K (x, x) 2(u (x)) T b(x) + (u (x)) T Au (x) = K (x, x) (u (x)) T b(x) = K (x, x) (u (x)) T Au (x). fasshauer@iit.edu MATH 590 Chapter 14 26

61 The Power Function However, by the definition of the cardinal functions Au (x) = b(x), and therefore we have the two new variants P K,X (x) = K (x, x) 2(u (x)) T b(x) + (u (x)) T Au (x) = K (x, x) (u (x)) T b(x) = K (x, x) (u (x)) T Au (x). Remark These formulas can be used for the numerical evaluation of the power function at x. fasshauer@iit.edu MATH 590 Chapter 14 26

62 The Power Function However, by the definition of the cardinal functions Au (x) = b(x), and therefore we have the two new variants P K,X (x) = K (x, x) 2(u (x)) T b(x) + (u (x)) T Au (x) = K (x, x) (u (x)) T b(x) = K (x, x) (u (x)) T Au (x). Remark These formulas can be used for the numerical evaluation of the power function at x. To this end one has to first find the value of the cardinal functions u (x) by solving the system Au (x) = b(x). fasshauer@iit.edu MATH 590 Chapter 14 26

63 The Power Function However, by the definition of the cardinal functions Au (x) = b(x), and therefore we have the two new variants P K,X (x) = K (x, x) 2(u (x)) T b(x) + (u (x)) T Au (x) = K (x, x) (u (x)) T b(x) = K (x, x) (u (x)) T Au (x). Remark These formulas can be used for the numerical evaluation of the power function at x. To this end one has to first find the value of the cardinal functions u (x) by solving the system Au (x) = b(x). This results in P K,X (x) = K (x, x) (b(x)) T A 1 b(x). (4) fasshauer@iit.edu MATH 590 Chapter 14 26

64 The Power Function Example Gaussian Power Function Figure: Data sites and power function for Gaussian interpolant with ε = 6 based on N = 81 uniformly gridded points in [0, 1] 2. fasshauer@iit.edu MATH 590 Chapter 14 27

65 The Power Function Example More Gaussian Power Function Figure: Data sites and power function for Gaussian interpolant with ε = 6 based on N = 81 tensor-product Chebyshev points in [0, 1] 2. fasshauer@iit.edu MATH 590 Chapter 14 28

66 The Power Function Example More Gaussian Power Function Figure: Data sites and power function for Gaussian interpolant with ε = 6 based on N = 81 Halton points in [0, 1] 2. fasshauer@iit.edu MATH 590 Chapter 14 29

67 The Power Function Remark Dependence of the power function on the data locations is clearly visible. MATH 590 Chapter 14 30

68 The Power Function Remark Dependence of the power function on the data locations is clearly visible. This connection was used in [De Marchi et al. (2005)] to iteratively obtain an optimal set of data locations that are independent of the data values. MATH 590 Chapter 14 30

69 The Power Function Remark Dependence of the power function on the data locations is clearly visible. This connection was used in [De Marchi et al. (2005)] to iteratively obtain an optimal set of data locations that are independent of the data values. Since A is a positive definite matrix whenever K is a strictly positive definite kernel we see that the power function satisfies the bounds 0 P K,X (x) = K (x, x) (u (x)) T Au (x) K (x, x). fasshauer@iit.edu MATH 590 Chapter 14 30

70 The Power Function Remark Dependence of the power function on the data locations is clearly visible. This connection was used in [De Marchi et al. (2005)] to iteratively obtain an optimal set of data locations that are independent of the data values. Since A is a positive definite matrix whenever K is a strictly positive definite kernel we see that the power function satisfies the bounds 0 P K,X (x) = K (x, x) (u (x)) T Au (x) K (x, x). At this point the power function is mostly a theoretical tool that helps us better understand error estimates since we can decouple the effects due to the data function f from those due to the kernel K and the data locations X (see the following theorem). fasshauer@iit.edu MATH 590 Chapter 14 30

71 The Power Function Remark Dependence of the power function on the data locations is clearly visible. This connection was used in [De Marchi et al. (2005)] to iteratively obtain an optimal set of data locations that are independent of the data values. Since A is a positive definite matrix whenever K is a strictly positive definite kernel we see that the power function satisfies the bounds 0 P K,X (x) = K (x, x) (u (x)) T Au (x) K (x, x). At this point the power function is mostly a theoretical tool that helps us better understand error estimates since we can decouple the effects due to the data function f from those due to the kernel K and the data locations X (see the following theorem). The power function is defined in an analogous way for strictly conditionally positive definite functions. fasshauer@iit.edu MATH 590 Chapter 14 30

72 Outline Generic Error Estimates for Functions in N K (Ω) 1 Fill Distance and Approximation Orders 2 Lagrange Form of the Interpolant and Cardinal Basis Functions 3 The Power Function 4 Generic Error Estimates for Functions in N K (Ω) 5 Error Estimates in Terms of the Fill Distance fasshauer@iit.edu MATH 590 Chapter 14 31

73 Generic Error Estimates for Functions in N K (Ω) Now we can give a first generic error estimate. fasshauer@iit.edu MATH 590 Chapter 14 32

74 Generic Error Estimates for Functions in N K (Ω) Now we can give a first generic error estimate. Theorem Let Ω R s, K C(Ω Ω) be strictly positive definite, and suppose that the points X = {x 1,..., x N } are distinct. Denote the interpolant to f N K (Ω) on X by P f. Then for every x Ω f (x) P f (x) P K,X (x) f NK (Ω). fasshauer@iit.edu MATH 590 Chapter 14 32

75 Generic Error Estimates for Functions in N K (Ω) Proof. Since f is assumed to lie in the native space of K the reproducing property of K yields f (x) = f, K (, x) NK (Ω). fasshauer@iit.edu MATH 590 Chapter 14 33

76 Generic Error Estimates for Functions in N K (Ω) Proof. Since f is assumed to lie in the native space of K the reproducing property of K yields f (x) = f, K (, x) NK (Ω). We express the interpolant in its cardinal form and apply the reproducing property of K. This gives us P f (x) = N f (x j )uj (x) j=1 fasshauer@iit.edu MATH 590 Chapter 14 33

77 Generic Error Estimates for Functions in N K (Ω) Proof. Since f is assumed to lie in the native space of K the reproducing property of K yields f (x) = f, K (, x) NK (Ω). We express the interpolant in its cardinal form and apply the reproducing property of K. This gives us P f (x) = = N f (x j )uj (x) j=1 N uj (x) f, K (, x j) NK (Ω) j=1 fasshauer@iit.edu MATH 590 Chapter 14 33

78 Generic Error Estimates for Functions in N K (Ω) Proof. Since f is assumed to lie in the native space of K the reproducing property of K yields f (x) = f, K (, x) NK (Ω). We express the interpolant in its cardinal form and apply the reproducing property of K. This gives us P f (x) = = N f (x j )uj (x) j=1 N uj (x) f, K (, x j) NK (Ω) j=1 = f, N uj (x)k (, x j) NK (Ω). j=1 fasshauer@iit.edu MATH 590 Chapter 14 33

79 Generic Error Estimates for Functions in N K (Ω) Proof (cont.) Now all that remains to be done is to combine the two formulas just derived and apply the Cauchy-Schwarz inequality. Thus, fasshauer@iit.edu MATH 590 Chapter 14 34

80 Generic Error Estimates for Functions in N K (Ω) Proof (cont.) Now all that remains to be done is to combine the two formulas just derived and apply the Cauchy-Schwarz inequality. Thus, N f (x) P f (x) = f, K (, x) uj (x)k (, x j) NK (Ω) j=1 fasshauer@iit.edu MATH 590 Chapter 14 34

81 Generic Error Estimates for Functions in N K (Ω) Proof (cont.) Now all that remains to be done is to combine the two formulas just derived and apply the Cauchy-Schwarz inequality. Thus, N f (x) P f (x) = f, K (, x) uj (x)k (, x j) NK (Ω) j=1 N f NK (Ω) K (, x) uj (x)k (, x j) j=1 NK (Ω) fasshauer@iit.edu MATH 590 Chapter 14 34

82 Generic Error Estimates for Functions in N K (Ω) Proof (cont.) Now all that remains to be done is to combine the two formulas just derived and apply the Cauchy-Schwarz inequality. Thus, N f (x) P f (x) = f, K (, x) uj (x)k (, x j) NK (Ω) j=1 N f NK (Ω) K (, x) uj (x)k (, x j) = f NK (Ω)P K,X (x), j=1 NK (Ω) where we have used the representation (2) of the quadratic form Q(u (x)) and the definition of the power function. fasshauer@iit.edu MATH 590 Chapter 14 34

83 Generic Error Estimates for Functions in N K (Ω) One of the main benefits of the above theorem is that we are now able to estimate the interpolation error by considering two independent phenomena: fasshauer@iit.edu MATH 590 Chapter 14 35

84 Generic Error Estimates for Functions in N K (Ω) One of the main benefits of the above theorem is that we are now able to estimate the interpolation error by considering two independent phenomena: the smoothness of the data (measured in terms of the native space norm of f which is independent of the data locations, but does depend on K ), fasshauer@iit.edu MATH 590 Chapter 14 35

85 Generic Error Estimates for Functions in N K (Ω) One of the main benefits of the above theorem is that we are now able to estimate the interpolation error by considering two independent phenomena: the smoothness of the data (measured in terms of the native space norm of f which is independent of the data locations, but does depend on K ), and the distribution of the data (measured in terms of the power function independent of the actual data values). fasshauer@iit.edu MATH 590 Chapter 14 35

86 Generic Error Estimates for Functions in N K (Ω) One of the main benefits of the above theorem is that we are now able to estimate the interpolation error by considering two independent phenomena: Remark the smoothness of the data (measured in terms of the native space norm of f which is independent of the data locations, but does depend on K ), and the distribution of the data (measured in terms of the power function independent of the actual data values). This is analogous to the standard error estimate for polynomial interpolation cited in most numerical analysis texts. fasshauer@iit.edu MATH 590 Chapter 14 35

87 Generic Error Estimates for Functions in N K (Ω) One of the main benefits of the above theorem is that we are now able to estimate the interpolation error by considering two independent phenomena: Remark the smoothness of the data (measured in terms of the native space norm of f which is independent of the data locations, but does depend on K ), and the distribution of the data (measured in terms of the power function independent of the actual data values). This is analogous to the standard error estimate for polynomial interpolation cited in most numerical analysis texts. Effects due to the use of any specific kernel K (or basic function in the translation invariant or radial case) are felt in both terms since the native space norm of f also varies with K. In particular, changing a possible shape parameter ε will have an effect on both terms in the error bound. fasshauer@iit.edu MATH 590 Chapter 14 35

88 Outline Error Estimates in Terms of the Fill Distance 1 Fill Distance and Approximation Orders 2 Lagrange Form of the Interpolant and Cardinal Basis Functions 3 The Power Function 4 Generic Error Estimates for Functions in N K (Ω) 5 Error Estimates in Terms of the Fill Distance fasshauer@iit.edu MATH 590 Chapter 14 36

89 Error Estimates in Terms of the Fill Distance The next steps are to refine this error bound by expressing the influence of the data locations in terms of the fill distance, MATH 590 Chapter 14 37

90 Error Estimates in Terms of the Fill Distance The next steps are to refine this error bound by expressing the influence of the data locations in terms of the fill distance, and the bound needs to be specialized to various choices of kernels K. MATH 590 Chapter 14 37

91 Error Estimates in Terms of the Fill Distance The next steps are to refine this error bound by expressing the influence of the data locations in terms of the fill distance, and the bound needs to be specialized to various choices of kernels K. Remark The most common strategy for obtaining error bounds in numerical analysis is to take advantage of the polynomial precision of a method (at least locally), and then to apply a Taylor expansion. fasshauer@iit.edu MATH 590 Chapter 14 37

92 Error Estimates in Terms of the Fill Distance With this in mind we observe Theorem Let Ω R s, and suppose K C(Ω Ω) is strictly positive definite. Let X = {x 1,..., x N } be a set of distinct points in Ω, and define the quadratic form Q(u) as in (2). The minimum of Q(u) is given for the vector u = u (x) of values of the cardinal functions, i.e., Q(u (x)) Q(u) for all u R N. fasshauer@iit.edu MATH 590 Chapter 14 38

93 Error Estimates in Terms of the Fill Distance Proof. We showed above (see (3)) that Q(u) = K (x, x) 2u T b(x) + u T Au. fasshauer@iit.edu MATH 590 Chapter 14 39

94 Error Estimates in Terms of the Fill Distance Proof. We showed above (see (3)) that Q(u) = K (x, x) 2u T b(x) + u T Au. The minimum of this quadratic form is given by the solution of the linear system Au = b(x). fasshauer@iit.edu MATH 590 Chapter 14 39

95 Error Estimates in Terms of the Fill Distance Proof. We showed above (see (3)) that Q(u) = K (x, x) 2u T b(x) + u T Au. The minimum of this quadratic form is given by the solution of the linear system Au = b(x). This, however, yields the cardinal functions u = u (x). fasshauer@iit.edu MATH 590 Chapter 14 39

96 Error Estimates in Terms of the Fill Distance In the proof below we will use a special coefficient vector ũ which provides the polynomial precision desired for the proof of the refined error estimate. fasshauer@iit.edu MATH 590 Chapter 14 40

97 Error Estimates in Terms of the Fill Distance In the proof below we will use a special coefficient vector ũ which provides the polynomial precision desired for the proof of the refined error estimate. Its existence is guaranteed by the following theorem on local polynomial reproduction proved in [Wendland (2005a)]. fasshauer@iit.edu MATH 590 Chapter 14 40

98 Error Estimates in Terms of the Fill Distance In the proof below we will use a special coefficient vector ũ which provides the polynomial precision desired for the proof of the refined error estimate. Its existence is guaranteed by the following theorem on local polynomial reproduction proved in [Wendland (2005a)]. The theorem requires Definition A region Ω R s satisfies an interior cone condition if there exists an angle θ (0, π/2) and a radius r > 0 such that for every x Ω there exists a unit vector ξ(x) such that the cone C = {x + λy : y R s, y 2 = 1, y T ξ(x) cos θ, λ [0, r]} is contained in Ω. fasshauer@iit.edu MATH 590 Chapter 14 40

99 Error Estimates in Terms of the Fill Distance The interior cone condition imposes a certain regularity on the domain Ω. In fact, a domain that satisfies this condition contains balls of a controllable radius. fasshauer@iit.edu MATH 590 Chapter 14 41

100 Error Estimates in Terms of the Fill Distance The interior cone condition imposes a certain regularity on the domain Ω. In fact, a domain that satisfies this condition contains balls of a controllable radius. In particular, this will be important when bounding the remainder of the Taylor expansions below. For more details see [Wendland (2005a)]. fasshauer@iit.edu MATH 590 Chapter 14 41

101 Error Estimates in Terms of the Fill Distance The interior cone condition imposes a certain regularity on the domain Ω. In fact, a domain that satisfies this condition contains balls of a controllable radius. In particular, this will be important when bounding the remainder of the Taylor expansions below. For more details see [Wendland (2005a)]. Existence of an approximation scheme with local polynomial precision is guaranteed by the following theorem. fasshauer@iit.edu MATH 590 Chapter 14 41

102 Error Estimates in Terms of the Fill Distance Theorem Suppose Ω R s is bounded and satisfies an interior cone condition, and let l be a non-negative integer. Then there exist positive constants h 0, c 1, and c 2 such that for all X = {x 1,..., x N } Ω with h X,Ω h 0 and every x Ω there exist numbers ũ 1 (x),..., ũ N (x) with N (1) ũ j (x)p(x j ) = p(x) for all polynomials p Π s l, (2) j=1 N ũ j (x) c 1, j=1 (3) ũ j (x) = 0 if x x j 2 c 2 h X,Ω. fasshauer@iit.edu MATH 590 Chapter 14 42

103 Error Estimates in Terms of the Fill Distance Theorem Suppose Ω R s is bounded and satisfies an interior cone condition, and let l be a non-negative integer. Then there exist positive constants h 0, c 1, and c 2 such that for all X = {x 1,..., x N } Ω with h X,Ω h 0 and every x Ω there exist numbers ũ 1 (x),..., ũ N (x) with N (1) ũ j (x)p(x j ) = p(x) for all polynomials p Π s l, (2) j=1 N ũ j (x) c 1, j=1 (3) ũ j (x) = 0 if x x j 2 c 2 h X,Ω. Property (1) yields the polynomial precision, and property (3) shows that the scheme is local. The bound in property (2) is essential for controlling the growth of error estimates. fasshauer@iit.edu MATH 590 Chapter 14 42

104 Error Estimates in Terms of the Fill Distance Theorem Suppose Ω R s is bounded and satisfies an interior cone condition, and let l be a non-negative integer. Then there exist positive constants h 0, c 1, and c 2 such that for all X = {x 1,..., x N } Ω with h X,Ω h 0 and every x Ω there exist numbers ũ 1 (x),..., ũ N (x) with N (1) ũ j (x)p(x j ) = p(x) for all polynomials p Π s l, (2) j=1 N ũ j (x) c 1, j=1 (3) ũ j (x) = 0 if x x j 2 c 2 h X,Ω. Property (1) yields the polynomial precision, and property (3) shows that the scheme is local. The bound in property (2) is essential for controlling the growth of error estimates. The quantity on the left-hand side of (2) is known as the Lebesgue constant at x. fasshauer@iit.edu MATH 590 Chapter 14 42

105 Error Estimates in Terms of the Fill Distance In the following theorem and its proof we will make repeated use of multi-index notation and multivariate Taylor expansions. MATH 590 Chapter 14 43

106 Error Estimates in Terms of the Fill Distance In the following theorem and its proof we will make repeated use of multi-index notation and multivariate Taylor expansions. For β = (β 1,..., β s ) N s 0 with β = s i=1 β i we define the differential operator D β as D β β = ( x 1 ) β 1 ( xs ). βs fasshauer@iit.edu MATH 590 Chapter 14 43

107 Error Estimates in Terms of the Fill Distance In the following theorem and its proof we will make repeated use of multi-index notation and multivariate Taylor expansions. For β = (β 1,..., β s ) N s 0 with β = s i=1 β i we define the differential operator D β as D β β = ( x 1 ) β 1 ( xs ). βs The notation D β 2 K (w, ) used below indicates that the operator is applied to K (w, ) viewed as a function of the second variable. fasshauer@iit.edu MATH 590 Chapter 14 43

108 Error Estimates in Terms of the Fill Distance The multivariate Taylor expansion of the function K (w, ) centered at w is given by with remainder K (w, z) = β <2k R(w, z) = D β 2 β =2k K (w, w) (z w) β + R(w, z) β! D β 2 K (w, ξ w,z) (z w) β, β! where ξ w,z lies somewhere on the line segment connecting w and z. fasshauer@iit.edu MATH 590 Chapter 14 44

109 Error Estimates in Terms of the Fill Distance Our earlier generic error estimate can now be formulated in terms of the fill distance. MATH 590 Chapter 14 45

110 Error Estimates in Terms of the Fill Distance Our earlier generic error estimate can now be formulated in terms of the fill distance. Theorem Suppose Ω R s is bounded and satisfies an interior cone condition. Suppose K C 2k (Ω Ω) is symmetric and strictly positive definite. Denote the interpolant to f N K (Ω) on the set X by P f. Then there exist positive constants h 0 and C (independent of x, f and K ) such that provided h X,Ω h 0. f (x) P f (x) Ch k X,Ω CK (x) f NK (Ω), fasshauer@iit.edu MATH 590 Chapter 14 45

111 Error Estimates in Terms of the Fill Distance Our earlier generic error estimate can now be formulated in terms of the fill distance. Theorem Suppose Ω R s is bounded and satisfies an interior cone condition. Suppose K C 2k (Ω Ω) is symmetric and strictly positive definite. Denote the interpolant to f N K (Ω) on the set X by P f. Then there exist positive constants h 0 and C (independent of x, f and K ) such that provided h X,Ω h 0. Here f (x) P f (x) Ch k X,Ω CK (x) f NK (Ω), C K (x) = max max β =2k w,z Ω B(x,c 2 h X,Ω ) Dβ 2 K (w, z) with B(x, c 2 h X,Ω ) denoting the ball of radius c 2 h X,Ω centered at x. fasshauer@iit.edu MATH 590 Chapter 14 45

112 Error Estimates in Terms of the Fill Distance Proof. By the generic error estimate in terms of the power function we know f (x) P f (x) P K,X (x) f NK (Ω). fasshauer@iit.edu MATH 590 Chapter 14 46

113 Error Estimates in Terms of the Fill Distance Proof. By the generic error estimate in terms of the power function we know f (x) P f (x) P K,X (x) f NK (Ω). Therefore, we now derive the bound P K,X (x) ChX,Ω k CK (x) for the power function in terms of the fill distance. fasshauer@iit.edu MATH 590 Chapter 14 46

114 Error Estimates in Terms of the Fill Distance Proof. By the generic error estimate in terms of the power function we know f (x) P f (x) P K,X (x) f NK (Ω). Therefore, we now derive the bound P K,X (x) Ch k X,Ω CK (x) for the power function in terms of the fill distance. We know that the power function is defined by [P K,X (x)] 2 = Q(u (x)). fasshauer@iit.edu MATH 590 Chapter 14 46

115 Error Estimates in Terms of the Fill Distance Proof. By the generic error estimate in terms of the power function we know f (x) P f (x) P K,X (x) f NK (Ω). Therefore, we now derive the bound P K,X (x) Ch k X,Ω CK (x) for the power function in terms of the fill distance. We know that the power function is defined by [P K,X (x)] 2 = Q(u (x)). Moreover, we know that the quadratic form Q(u) is minimized by u = u (x). fasshauer@iit.edu MATH 590 Chapter 14 46

116 Error Estimates in Terms of the Fill Distance Proof. By the generic error estimate in terms of the power function we know f (x) P f (x) P K,X (x) f NK (Ω). Therefore, we now derive the bound P K,X (x) Ch k X,Ω CK (x) for the power function in terms of the fill distance. We know that the power function is defined by [P K,X (x)] 2 = Q(u (x)). Moreover, we know that the quadratic form Q(u) is minimized by u = u (x). Therefore, any other coefficient vector u will yield an upper bound on the power function. fasshauer@iit.edu MATH 590 Chapter 14 46

117 Error Estimates in Terms of the Fill Distance We take u = ũ(x) from the theorem guaranteeing existence of a local polynomial reproduction so that we are ensured to have polynomial precision of degree l 2k 1. fasshauer@iit.edu MATH 590 Chapter 14 47

118 Error Estimates in Terms of the Fill Distance We take u = ũ(x) from the theorem guaranteeing existence of a local polynomial reproduction so that we are ensured to have polynomial precision of degree l 2k 1. For this specific choice of coefficients we have [P K,X (x)] 2 Q(u) = K (x, x) 2 u j K (x, x j ) + u i u j K (x i, x j ), j i j where the sums are over those indices j with u j 0. fasshauer@iit.edu MATH 590 Chapter 14 47

119 Error Estimates in Terms of the Fill Distance We take u = ũ(x) from the theorem guaranteeing existence of a local polynomial reproduction so that we are ensured to have polynomial precision of degree l 2k 1. For this specific choice of coefficients we have [P K,X (x)] 2 Q(u) = K (x, x) 2 u j K (x, x j ) + u i u j K (x i, x j ), j i j where the sums are over those indices j with u j 0. Now we apply the Taylor expansion centered at x to K (x, ) and centered at x i to K (x i, ), and evaluate both functions at x j. fasshauer@iit.edu MATH 590 Chapter 14 47

120 Error Estimates in Terms of the Fill Distance We take u = ũ(x) from the theorem guaranteeing existence of a local polynomial reproduction so that we are ensured to have polynomial precision of degree l 2k 1. For this specific choice of coefficients we have [P K,X (x)] 2 Q(u) = K (x, x) 2 u j K (x, x j ) + u i u j K (x i, x j ), j i j where the sums are over those indices j with u j 0. Now we apply the Taylor expansion centered at x to K (x, ) and centered at x i to K (x i, ), and evaluate both functions at x j. This yields Q(u) = K (x, x) 2 u j D β 2 K (x, x) (x j x) β + R(x, x j ) β! j β <2k + u i u j D β 2 K (x i, x i ) (x j x i ) β + R(x i, x j ). β! i j β <2k fasshauer@iit.edu MATH 590 Chapter 14 47

121 Error Estimates in Terms of the Fill Distance Next, we identify p(z) = (z x) β so that p(x) = 0 unless β = 0. fasshauer@iit.edu MATH 590 Chapter 14 48

122 Error Estimates in Terms of the Fill Distance Next, we identify p(z) = (z x) β so that p(x) = 0 unless β = 0. Therefore the polynomial precision property of the coefficient vector u simplifies this expression to Q(u) = K (x, x) 2K (x, x) 2 j u j R(x, x j ) + i u i β <2k D β 2 K (x i, x i ) (x x i ) β + β! i j u i u j R(x i, x(5) j ). fasshauer@iit.edu MATH 590 Chapter 14 48

123 Error Estimates in Terms of the Fill Distance Next, we identify p(z) = (z x) β so that p(x) = 0 unless β = 0. Therefore the polynomial precision property of the coefficient vector u simplifies this expression to Q(u) = K (x, x) 2K (x, x) 2 j u j R(x, x j ) + i u i β <2k D β 2 K (x i, x i ) (x x i ) β + β! i j u i u j R(x i, x(5) j ). Now we can apply the Taylor expansion again and make the observation that β <2k D β 2 K (x i, x i ) (x x i ) β = K (x i, x) R(x i, x). (6) β! fasshauer@iit.edu MATH 590 Chapter 14 48

124 Error Estimates in Terms of the Fill Distance If we use (6) and rearrange the terms in (5) we get Q(u) = K (x, x) [ u j 2R(x, x j ) j i u i R(x i, x j ) ] + i u i [K (x i, x) R(x i, x)]. (7) fasshauer@iit.edu MATH 590 Chapter 14 49

125 Error Estimates in Terms of the Fill Distance If we use (6) and rearrange the terms in (5) we get Q(u) = K (x, x) [ u j 2R(x, x j ) j i u i R(x i, x j ) ] + i u i [K (x i, x) R(x i, x)]. (7) One final Taylor expansion we need is (using the symmetry of K ) K (x i, x) = K (x, x i ) = β <2k D β 2 K (x, x) (x i x) β + R(x, x i ). (8) β! fasshauer@iit.edu MATH 590 Chapter 14 49

126 Error Estimates in Terms of the Fill Distance If we use (6) and rearrange the terms in (5) we get Q(u) = K (x, x) [ u j 2R(x, x j ) j i u i R(x i, x j ) ] + i u i [K (x i, x) R(x i, x)]. (7) One final Taylor expansion we need is (using the symmetry of K ) K (x i, x) = K (x, x i ) = β <2k D β 2 K (x, x) (x i x) β + R(x, x i ). (8) β! If we insert (8) into (7) and once more take advantage of the polynomial precision property of the coefficient vector u we are left with Q(u) = [ u j R(x, x j ) + R(x j, x) ] u i R(x i, x j ). j i fasshauer@iit.edu MATH 590 Chapter 14 49

127 Error Estimates in Terms of the Fill Distance From the previous slide we have Q(u) = [ u j R(x, x j ) + R(x j, x) j i u i R(x i, x j ) ]. fasshauer@iit.edu MATH 590 Chapter 14 50

128 Error Estimates in Terms of the Fill Distance From the previous slide we have Q(u) = [ u j R(x, x j ) + R(x j, x) j i u i R(x i, x j ) ]. Now the theorem on local polynomial reproductions allows us to bound j u j c 1. fasshauer@iit.edu MATH 590 Chapter 14 50

129 Error Estimates in Terms of the Fill Distance From the previous slide we have Q(u) = [ u j R(x, x j ) + R(x j, x) j i u i R(x i, x j ) ]. Now the theorem on local polynomial reproductions allows us to bound j u j c 1. Moreover, x x j 2 c 2 h X,Ω and x i x j 2 2c 2 h X,Ω. fasshauer@iit.edu MATH 590 Chapter 14 50